| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s − 3·9-s + 2·10-s − 2·11-s + 3·13-s − 14-s + 16-s + 3·17-s − 3·18-s + 4·19-s + 2·20-s − 2·22-s + 23-s − 25-s + 3·26-s − 28-s − 7·31-s + 32-s + 3·34-s − 2·35-s − 3·36-s + 6·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.632·10-s − 0.603·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.707·18-s + 0.917·19-s + 0.447·20-s − 0.426·22-s + 0.208·23-s − 1/5·25-s + 0.588·26-s − 0.188·28-s − 1.25·31-s + 0.176·32-s + 0.514·34-s − 0.338·35-s − 1/2·36-s + 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.044306304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.044306304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.92413644141020, −12.45482113756791, −11.77928534538152, −11.52843464253575, −10.91267799112353, −10.66930572044203, −9.966433359967410, −9.546177037503212, −9.267240101180388, −8.542445939155244, −8.015151378867320, −7.704251765194126, −6.995099005643160, −6.444865209386614, −6.000286894405393, −5.674323010092715, −5.190563776753541, −4.864679694112725, −3.901608588295147, −3.471584035250343, −3.023344013921037, −2.539275009152945, −1.850707533893096, −1.312845461157105, −0.4616729225191263,
0.4616729225191263, 1.312845461157105, 1.850707533893096, 2.539275009152945, 3.023344013921037, 3.471584035250343, 3.901608588295147, 4.864679694112725, 5.190563776753541, 5.674323010092715, 6.000286894405393, 6.444865209386614, 6.995099005643160, 7.704251765194126, 8.015151378867320, 8.542445939155244, 9.267240101180388, 9.546177037503212, 9.966433359967410, 10.66930572044203, 10.91267799112353, 11.52843464253575, 11.77928534538152, 12.45482113756791, 12.92413644141020
